The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains
We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to main...
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Format | Journal Article |
Language | English |
Published |
29.01.2024
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Abstract | We consider the $h$-version of the finite-element method, where accuracy is
increased by decreasing the meshwidth $h$ while keeping the polynomial degree
$p$ constant, applied to the Helmholtz equation. Although the question "how
quickly must $h$ decrease as the wavenumber $k$ increases to maintain
accuracy?" has been studied intensively since the 1990s, none of the existing
rigorous wavenumber-explicit analyses take into account the approximation of
the geometry. In this paper we prove that for nontrapping problems solved using
straight elements the geometric error is order $kh$, which is then less than
the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then
illustrated in numerical experiments. More generally, we prove that, even for
problems with strong trapping, using degree four (in 2-d) or degree five (in
3-d) polynomials and isoparametric elements ensures that the geometric error is
smaller than the pollution error for most large wavenumbers. |
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AbstractList | We consider the $h$-version of the finite-element method, where accuracy is
increased by decreasing the meshwidth $h$ while keeping the polynomial degree
$p$ constant, applied to the Helmholtz equation. Although the question "how
quickly must $h$ decrease as the wavenumber $k$ increases to maintain
accuracy?" has been studied intensively since the 1990s, none of the existing
rigorous wavenumber-explicit analyses take into account the approximation of
the geometry. In this paper we prove that for nontrapping problems solved using
straight elements the geometric error is order $kh$, which is then less than
the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then
illustrated in numerical experiments. More generally, we prove that, even for
problems with strong trapping, using degree four (in 2-d) or degree five (in
3-d) polynomials and isoparametric elements ensures that the geometric error is
smaller than the pollution error for most large wavenumbers. |
Author | Chaumont-Frelet, Théophile Spence, Euan A |
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BackLink | https://doi.org/10.48550/arXiv.2401.16413$$DView paper in arXiv |
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Snippet | We consider the $h$-version of the finite-element method, where accuracy is
increased by decreasing the meshwidth $h$ while keeping the polynomial degree
$p$... |
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SubjectTerms | Computer Science - Numerical Analysis Mathematics - Numerical Analysis |
Title | The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains |
URI | https://arxiv.org/abs/2401.16413 |
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